We now show (). Using [1]} and (REF ), it holds that for any \(\beta \in \left(0,\beta _{\alpha ,q}\right)\) ,
\(\mathbb {P}\left( \theta _n \ge \mathrm {e}^{-\alpha \beta n } \right) \ge \exp (-n \mathrm {KL}(q_A-x_{\alpha ,\beta } | q_A) - 2^{-1} \log (n)) \;,\) 
